In Partial Fulfillment of
the Requirements for the
Bachelor of Fine Arts Degree

by
Tom Lechner
May 10, 1998

The Year in Review

I had stated in my proposal that I would most likely be making several prints, and not doing any sculpture.
Really, I wanted to continue in my traditional work ways, which have been making prints and also making sculpture.
Each area has a very different focus for me, and I saw no way to adequately merge the two, as my committee suggested.
My sculpture has almost exclusively been slanted toward analytical mathematical exploration, by way of woodworking.
My prints tend to have a much more direct relation to how I see the social human world, and nothing to do with math.

I finally decided I would make a large painting in the manner of my prints, and also a companion wooden mathematical object.
They would be related superficially by using similar colors, and similar compositions. This could be done since this particular
math shape has a very intricate exterior, and it is easier to find relations between things that are fairly complicated.
Elements in the painting would associate to each other in a similar way as in the shape, in that areas of detail would
dip under each other, change color, grow larger, and so on. These elements would be slices of the life of a hapless loner,
whose only real stability is to study technical things, such as the math shape at hand. He is not exactly a loner, because
he has several complicated relationships, all of which seem to break apart for no apparent reason. Things inevitably
fracture or twist away, but there is still a kind of order to it, just like the objects of his fascination.

As I had first conceived it, the painting and the sculpture could exist just fine without each other. They would certainly
be related, but more like cousins living on opposite sides of a city, rather than as inseparable passionate partners, who
will die without each other. The painting is wrapped up in itself, and the object presents itself boldly, without regard
for the painting. This parallels the life of the man, who keeps repeating the same mistakes, but each time from a different
angle. The objects he likes have an angle on everything, and they are always the correct angles, but really, it's nothing
to do with him. He knows it, too, the poor fool.

To treat that relationship fully would probably take too long for me to complete in the one semester especially after
being sick for a week, so I modified my plans. I decided to focus on the wooden math object. This meant make one, and if
there was time, make two. I would want to make two first to show that the first one wasn't a mistake, and second because
this particular shape is structurally identical to its mirror image, but has an opposite twist. I would make the mirror image.

My committee suggested I take my notes with the calculations and diagrams that I used to figure out and construct the object,
and put those on the wall. As a bare bones fail safe tactic to have more stuff, hanging my notes is perhaps acceptable, but
I was not very thrilled by the idea. I have seen some work where such notes are presented as an art form, perhaps to make
some obscure point about mysterious arcane knowledge or simply to generate snazzy wallpaper. I may like to make a point again
and again, but not that point. As interesting as I may find trying to figure out diagrams that don't make sense to me, my notes
do make sense to me, and I much prefer to present them so that they potentially can be understood. This basically means reworking
my notes so that they become something that I would learn well from, if I happened to come across it. Probably no one will take
the time and no one will care, but I feel it is a much stronger presentation to show how I learn and think about things, rather
than show fragments of what I have accomplished. My raw notes are a fragment of how I deal with the shape. The reworking is
also that, but it is more complete.

The Final Form

I have been able to complete one Great Inverted Retrosnub Icosidodecahedron, which I will henceforth refer to as the Giri.
I have also made the basic skeleton of its mirror image, which if complete would have the opposite color scheme as the
original. Also, my notes have become a large hexagonal diagram showing how the 12 pentagrams and 80 triangles chop each
other, plus a number of smaller diagrams around the edges of this which correspond to each visible type of piece of the
completed shape. Beneath these diagrams are brief composition notes. My original notes I have assembled into a booklet of
supplementary notes on a little shelf below the display. No matter how much I can describe this object there are going to
be things that one must try to see, and a supplementary notebook helps to fill that need. A few feet from the display is
the incomplete Giri, which I have dubbed malnourished, and a few feet more from the display is the completed form.

When I saw there was no chance of completing both Giris, I had hoped I would be able to add enough onto the second, so that
half would be a complete surface and would face its completed brother, and the other half would just be the basic skeleton
which can be seen through and which faces the diagrams on the wall. There would not be an abrupt transition so that it would
be apparent which pieces must be added and in what order to go from the skeleton to completion. This form of partial completion
makes a better show than the second piece being all in one state, but probably any partial form can act as a good transition
from the diagrams to the complete form.

Sure, it's pretty, but so what?

Such is the sentiment of some smug people when confronted by such cold intellectual things as math, or sculpture that embraces
primarily mathematical ideas. Such things are certainly not art that does its darnedest to establish the dominance of ambiguity
and random chance. It is art that values patience and perseverance, not simply to amaze people with feats of mental endurance,
but to comprehend basic forms of nature. Math is every bit as passionate, challenging, and profound as anything modern art critics
exalt. It is foolish to dismiss an activity on the grounds that it is nothing but technical. Even if it is merely technical, I
can at least say that I get off on a technicality.

Whether math forms are indeed basic forms of nature to anyone who doesn't study math or science is perhaps debatable, mainly
because math does not directly address human emotions. There's more to life than being swept away, just as there's more to
life than cold calculation.

Math and Art

There was a movement in art of this century to concentrate on primary forms, for instance gesture and simple lines or basic
color interactions or some prevalent mood of the artist in abstract expressionism. In minimalism, artists pined for basic forms
and the best they could do was usually a square, or cube. This is where minimalism failed to do anything very interesting,
except perhaps to art critics. One might find the universe in a six foot blob of color, but then again, one might not. If
these artists had a better sense of geometry and math, they could have made more interesting structures, but perhaps banality
was a main point of abstract art. I am surely demonstrating selective attention. Anyone who spends their life making art of
some kind usually does not make dismal art every time, even if they do come from New York.

I see my work a little like an extension of minimalism, with certain parallels to architecture. The cult of the square affected
modern architecture also, notably through such architects as Mies Van der Roe and Philip Johnson for whom the pure form of a
building is very important. A statement of pure form is fine and good, but people are going to use this form, and architects
need satisfied clients, though they'll do their best to ignore that. An architect must consider the little things that run
through it, and worry about them making a mess and rearranging the furniture, thereby tarnishing the pure form. Architects
are artists who want a large number of people to work for them. Building buildings requires extensive preliminary planning in
terms of finding materials, figuring out how they go together, and finally putting those materials together properly. These
last concerns, as well as the pursuit of a beautiful structure just to have a beautiful structure are related to how I
approach my sculptures.

Sometimes people suggest I would be a great architect because of my love of form and perfectionist methodical ways. I am not
going to argue with that, but when I look back over my short life, the things that have most moved me aside from people, are
all some form of fine art, and not architecture. Perhaps over time as I become a more seasoned world traveler, and see more
of the potential of various kinds of activity, my goals will change. For now, however, fine art as I understand it is the
area that offers me the most potential for satisfaction.

Scientists study forms that already exist in nature, and invent other forms that mimic nature in some small way. How well
a scientific theory is accepted depends on the types of observations scientists are able to make. Western scientific
theories for the past few hundred years have tended to have a considerable mathematical component. This component is
the most compact way scientists can record how they can mimic nature. In the oft repeated words of the eminent physicist
P.A.M. Dirac, "A physical law must possess mathematical beauty." Mathematicians are the true minimalists, in that they study
very abstract forms, that to the casual observer have nothing to do with anything. Dirac also said, "The mathematician plays
a game in which he himself invents the rules while the physicist plays a game in which the rules are provided by Nature, but
as time goes on it becomes increasingly evident that the rules which the mathematician finds interesting are the same as those
which Nature has chosen."

I have used math in my art, but my ultimate goals are more related to my visual aesthetics. I prefer to make things that I can
look at, rather than make things I can only think about. Maybe if I thought more clearly, my aims would be different, but perhaps
it just means I'm better suited to be an artist than a mathematician. I was pleased to discover the words of the eminent geometer
H.S.M. Coxeter, who said "Thus, the chief reason for studying regular polyhedra is still the same as in the time of the
Pythagoreans, namely, that their symmetrical shapes appeal to one's artistic sense."

People have been captivated by patterns and ways in which abstract things can interact for a long time. Perhaps the most
notably early example in recorded history is the Pythagorean Brotherhood. Not much is firmly known about Pythagoras, but
it is known that he espoused the idea that one can study math for its intrinsic interest. He did not have many followers
at first, and in fact had to bribe his first pupil. Over time, Pythagoras had less funds available and reduced and eventually
curtailed his bribe, but his pupil kept on from shear interest. Over time, he converted many followers, who were all sworn to
secrecy about the ideas developed. In fact, after his death, one former follower was drowned by other followers for announcing
the discovery of the Dodecahedron. Pythagoras and many of his followers were murdered by an uprising of a hate monger's devotees,
who condemned the Brotherhood for its secrecy, which meant they were ruining life for everybody else. Math has its share of
adventure.

In Islamic culture many years ago, symmetric patterns were the decorative element in temples. No guys with wings or flowing
robes for them! The magic of the patterns was the only appropriate form of display. Any portrayal was seen as a mere copy of
the works of Creation, and could only be caricature, which is wrong in the eyes of God, "and on the last day, the wretched
image makers will be required to blow life into their creations." So far, my sculpture and I are safe on that point. Islamic
patterns greatly influenced the graphic artist M.C. Escher, who later went on to incorporate recognizable figures into those
and other patterns. He had a long and industrious career of making prints that blended his interest in people and nature,
particularly symmetric patterns. When talking about the math in his work, he once wrote, "Crystallographers have put forward
a definition of the idea, they have ascertained which and how many systems or ways there are of dividing a plane in a regular
manner. In doing so, they have opened the gate leading to an extensive domain, but they have not entered this domain themselves.
By their very nature, they are more interested in the way in which the gate is opened than in the garden lying behind it." One
might say that my thesis work represents me still examining the gate. I don't think I would say that, but one might.

Particular interests

The many geometric shapes I have made in the last four years are mostly of wood. One is plastic and a couple are metal.
Sometimes I make paper models primarily to get more of an idea of how a shape is structured. I prefer wood. The other
materials are all pretty much uniform. For shapes that have several hundred pieces to cut out, this for me is a drawback.
Wood at every step holds me more.

In making my wooden shapes, I first find some form that strikes me somehow. My thesis shape, the Giri, I first found in a
book by Magnus J. Wenninger, who has been studying these math shapes for over 50 years. He has published three books with
about a hundred different very complicated shapes, all made in paper. The Giri struck me because of its relatively simple
composition of 12 pentagrams and 80 triangles, and also because it looked like more than just a big mess at first glance.
There is great regularity to it, but it is rich enough that it is not entirely apparent at first, but once one does start
to follow a line, one sees that it goes all the way to the other side to another point, then turns around and comes all the
way back, then all the way away, and so on forever. I find particularly amazing the pits that look somewhat like flowers,
which I have made a red wood of chokte kok smartwood sitting down in a very dark wood of peruvian walnut.

I recently found the email address of Magnus Wenninger (born in 1919) and told him of my thesis. He said it is unusual to
hear of anyone making this particular shape, and astounding since he knows what is involved. And in wood! Even more terrific.

Once I decide on some shape that really intrigues me, the next step is to work out how it shall look. This involves determining
what aspects of the shape to highlight, such as particular symmetries, or the above mentioned pits. Then I must decide what woods
to use, what holes to leave if any, what size, and how to manipulate the grain of the wood. Sometimes while investigating shapes,
particular woods will stand out as being excellent for particular shapes while I browse the wood store, or a shape will suggest
certain wood combinations.

I don't normally work out completely beforehand how a shape will look in the end, because generally I find it easier to make
something if I don't completely know how it will turn out. Wood is harder to predict than paper or metal or plastic. Those
other things are all manufactured or refined considerably, and so one piece looks a lot like the next piece. Unpredictability
with those materials is mostly how the material stretches, such as metal with heat. Each piece of wood comes from a particular
tree, and that particular quality comes through in the grain, from the coloring to grain density to grain patterning, to how
pieces will react to various finishes, and how pieces will respond to tools. It takes a long time to become well versed with
how woods behave, and I am only slightly versed. I am also interested in how wood behaves before it is cut up, specific factors
that govern how a tree grows and appears , and what trees are cut down and where, but I have not yet invested much time to
that end.

In the Giri, I have used 7 different woods: peruvian walnut for the deep dark pits, chokte kok smartwood for the turbine
like pit centers, bloodwood for the little sliver resting on the maple, maple, which I chose for this piece's wavy nature
because it runs around the pits like a river valley, jatoba for one fjord wall of the river valley, spanish cedar for the
adjacent wall, and beech for the part of the pentagrams bordering the pits. Perhaps there is some deeper meaning for my
choice of imagery to associate to each type of piece, but I tend to consider my imagery here more a form of note keeping,
as well as a source of amusement, on par with finding patterns in ink blots and clouds. Leonardo da Vinci recommended that
sort of thing to help in creating ideas for compositions. Salvador Dali or Buckminster Fuller might really fly with the
pursuit of the implications of what patterns they see, and perhaps I will too in time, but I seem to have gotten tangled
in Escher's gate. I somewhat see beyond, but I am perhaps not holistically engaged. I do not agree with that assessment,
but then I am too close to the problem to see clearly.

Once I know what woods I want and how things will go together, then I must figure out precisely the shape of each piece,
and devise jigs to help me make those shapes. This is the most death defying stage, because I typically use power tools
extensively and the table saw in particular. I try to be very safe, because I value my fingers, and no, you cannot examine
my hands closely. Over the years I have developed some sense of what will hold a piece through the saw and what won't and
what to expect after a cut, specifically how excess pieces fall. For instance what makes a piece dangerously shoot from the
saw or on what a blade is likely to bind. It is very difficult to predict everything, but experience certainly helps. See
the Construction appendix for more detailed jig information.

After cutting everything out, they must be glued together. This introduces many more deterrents to making such things. I
typically use ordinary carpenter's yellow glue. which takes about a minute to stick, about half an hour to set well, and
many hours to dry. I normally build up a shape in chunks or rings then attach chunks or fill in the rings, as that helps
spread errors around, rather than concentrating all my past mistakes into an ultimate moment. Each joint must be pressed
firmly for about half a minute then left alone to set. For some pieces I cut grooves along the seams, then put little slabs
of wood in the grooves, which helps the pieces stay together while gluing, and also provides additional mechanical strength
for the glue joint.

Eventually, the thing is all glued together, and the penultimate stage is cleaning things up. This entails final sanding and
filling any unsightly gaps. I always use little slivers of wood to fill gaps because it disappears into the wood more than
putty, and putty is too uniform anyway. Most of the sanding of individual faces I do before gluing, because it is so much
easier. Sometimes my fingers with glue on them touch where they shouldn't, or I accidentally drop something on it, and
sanding those blemishes prepares the wood to take a finish evenly. I'll normally sand the edges so people can run their
fingers across them without cutting themselves and without getting splinters.

If all these stages only leave me craving more, then the final step is to put a finish on the piece. I usually use a 1
to 1 to 1 mixture of boiled linseed oil, turpentine, and urethane. This tends to yellow and darken the wood slightly,
and adds a little more unpredictability to the whole endeavor. As time goes on of course, I learn what I can expect this
finish to do. Any little excess glue spots brashly proclaim their existence when the finish is applied and can at this
time be once and for all obliterated with sandpaper and a fresh application of finish. I normally let this finish soak
in for about 20 minutes, then wipe off any excess with a soft rag, because any excess at this point will most likely not
be absorbed by the wood, and if left, will only turn forever sticky. When the finish dries, I smooth it down a little with
steel wool. With some shapes it is easier to finish the pieces before they are glued, for instance when a shape has visible
interior structure. The Giri has a very complicated external structure, and I used a little butter knife shaped piece of
wood with cloth around it to help remove any excess finish that gathers around the corners.

They don't call it a finish for nothing, and it is at this time that I collapse from such an amazing feat of mental and
physical endurance, only to arise shortly after to make another one while I still have a full wood shop at my disposal.

Appendix A: Construction

This section describes specific jigs and wood
construction information in general and also
specifically about the
Great Inverted Retrosnub Icosidodecahedron.

(Insert lots of diagrams here, they're in the printed book, maybe I'll post them on the site when I'm not lazy)

Appendix B: Math

This section contains some of the
mathematics relating to the Great Inverted
Retrosnub Icosidodecahedron.